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TIME DOMAIN SCATTERING AND 
RADAR CROSS SECTION CALCULATIONS 
FOR A THIN, COATED PERFECTLY 
CONDUCTING PLATE 


furnished to 

Captain Doug Havens 
AFEWC/ESAS 

San Antonio, TX 78243-5000 
and to 

Dr. Randy Jost 
AFWAL/CDJ 

Wright-Patterson AFB, OH 45433-6523 


submitted by 

Raymond J. Luebbers and John H. Beggs 
Electrical and Computer Engineering Department 
The Pennsylvania State University 
University Park, PA 16802 

(814) 865-2362 


February 1991 


(NASA-CR-190106) TIME DOMAIN SCATTERING AND N92-26114 

RADAR CROSS SFCTION CALCULATIONS FOR A THIN, 

COATED PERFECTLY CONDUCTING PLATE 
(Pennsylvania State Univ.) 10 p 


Unci as 

G3/32 0077913 


PRECEDING PAGE BLANK NOT FILMED 

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Abstract 

Radar Cross Section (RCS) calculations for flat, perfectly conducting plates are 
readily available through the use of conventional frequency domain techniques such as 
the Method of Moments (MOM). However, if the plate is covered with a dielectric material 
that is relatively thick in comparison with the wavelength in the material, these frequency 
domain techniques become increasingly difficult to apply. In this paper , we present the 
application of the Finite Difference Time Domain (FDTD) technique to the problem of 
electromagnetic scattering and RCS calculations from a thin, perfectly conducting plate 
that is coated with a thick layer of lossless dielectric material. Both time domain and RCS 
calculations will be presented and discussed. 

I. Introduction 

The Finite Difference Time Domain (FDTD) technique has become increasingly 
popular in recent years for modeling electromagnetic scattering problems. It is based 
upon the time domain form of Maxwell’s equations, in which temporal and spatial 
derivatives are approximated by finite differences, and the electric and magnetic fields are 
interleaved spatially and temporally. Transient scattering behavior is easily examined and 
through the use of non-sinusoidal plane wave excitation, wideband frequency resuits can 
be obtained. The technique was first proposed by Yee [1] in 1966 and is inherently 
volumetric, which makes it ideal for modeling volumetric scatterers. Thin scatterers can 
easily be accommodated, and recently the technique has been expanded to include 
dispersive materials [2], plasmas [3], and chiral materials [4]. Through the use of a near 
to far zone transformation [5], far zone scattered fields (and thus RCS data) are readily 
available. This paper presents time domain scattering and RCS calculations over 0-3 GHz 
for several incidence angles from a thin, perfectly conducting (PEC) plate that is coated 
with a uniform lossless dielectric layer. 

II. Problem Description 

The scattering problem was a 3A. by 6X (at 3 GHz) perfectly conducting plate that 
was coated with a 5 cm thick lossless dielectric layer with relative dielectric constant of 
c r =4.0. Figure 1 shows the problem geometry with the dielectric layer on top of the plate. 

The wavelength (at 3 GHz) inside the dielectric layer is = 5 -° cm - where is the 

free space wavelength at 3 GHz. Thus, the dielectric coating is relatively thick at 1/.. The 
spatial increment (cell size) was chosen to be 1 cm, which provides a spatial resolution 
of 5 cells/A. inside the dielectric coating and 10 cells/X 0 in free space. 

The problem space size was chosen to be 61 by 121 by 49 cells in the x. y and 
z directions respectively. The plate was centered within the problem space in the x and 
y directions. The plate was positioned low in the problem space in the z directicn 


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to allow any specular reflections multiple encounters with the outer radiation boundary 
condition (ORBC). 

The plate was constructed with 30 by 60 by 5 cells in the x, y and z directions for 
the dielectric coating, and with 30 by 60 by 0 cells for the PEC plate. The dielectric 
coating was constructed first, and the PEC plate was constructed on the bottom of the 
dielectric layer to avoid any air gaps within the scatterer. A 15 cell and 30 cell border on 
each side of the scatterer in the x and y directions provided adequate margin for the near 
to far zone transformation integration surface and for the ORBC. 

A 0 -polarized, Gaussian pulse incident plane wave with a maximum amplitude of 
1000 V/m and a total temporal width of 128 time steps was chosen. The time step was 
0.0192 ns and the total number of time steps was 2048. 

III. Computations and Discussion 

Calculations were made at incidence angles 0 =0.0, 0 =60.0, 0 =85.0 and 0 =90.0 
degrees for both an uncoated and coated plate. The incidence angle was taken from the 
+z axis, the <p incidence angle was <p =0.0 degrees for all computations, and the far field 
computations were for backscatter only. The ©-polarized scattered and incident fields 
were then transformed to the frequency domain via an FFT and RCS was determined. 
Each computational problem required slightly more than one hour of CPU time on a Cray 
Y-MP supercomputer. 

Figure 2 shows the 0 -polarized time domain far zone scattered electric field for 
incidence angle 0 =0.0 degrees. Note for the coated plate the early reflection from the 
top edge of the dielectric layer. Also note the time domain response for the coated plate 
is not markedly different from the uncoated plate except for some additional ringing due 
to energy being confined within the dielectric coating. Figure 3 shows the RCS 
computations versus frequency again for both the uncoated and coated plate, and the 
RCS for both cases does not differ substantially. 

Figure 4 shows the 0 -polarized time domain far zone scattered field for incidence 
angle 0 =60.0 degrees. Note the time responses differ more substantially for this case 
as more energy is being confined within surface wave modes of the dielectric layer. 
Figure 5 shows the corresponding RCS. Note the small peaks that have appeared in the 
RCS for the coated plate. We postulate these peaks correspond to surface wave modes 
that have been excited and radiate energy to the far field. As an approximation, we 
computed cutoff frequencies for waveguide modes for an infinite, dielectric covered 
ground plane according to Balanis [6]. These waveguide modes and corresponding 
cutoff frequencies are tabulated in Table 1. Examining the RCS of the coated plate, it is 
easily seen that a peak in the RCS is in close proximity to each cutoff frequency from 
Table 1 . The peaks in the RCS are not located exactly at the cutoff frequencies of Table 
1 , probably due to the finite size of the plate. 


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Mode 

Cutoff Frequency (GHz) 

TM 0 

0.00000 

TE, 

0.86589 

tm 2 

1.73718 

te 3 

2.59767 


Table 1 . Modes and cutoff frequencies for 
an infinite ground plane covered with a 
5 cm thick dielectric layer of e r =4.0. 

Figure 6 shows the 0 -polarized time domain far zone scattered field for incidence 
angle 9 =85.0 degrees. Also shown in Figure 6 is the far zone scattered field for a 5 cm 
thick 3k by 6k dielectric layer. Since this incidence angle is near grazing, we expect to 
see little scattered field for the uncoated plate. Examining the time responses in Figure 
6, the uncoated plate response is indeed quite small in comparison to the dominant 
coated plate response. The dielectric layer response has the same general form as the 
coated plate response but is smaller in magnitude. Figure 7 shows the corresponding 
RCS. Note the large peaks and lobing structure for the coated plate and dielectric layer 
RCS. These can be attributed to radiation from surface wave modes of the dielectric 
cavity. 


Figure 8 shows the 0 -polarized time domain far zone scattered field for incidence 
angle© =90.0 degrees. The response for the zero-thickness uncoated plate is zero, while 
the coated plate time response does not differ substantially from that for incidence angle 
0 =85.0 degrees. Figure 9 shows the RCS for the coated plate only, and it is also similar 
to the coated plate RCS for incidence angle 0 =85.0 degrees. 

IV. Conclusions 

In this paper, the FDTD technique has been applied to model electromagnetic 
scattering from a perfectly conducting plate coated with a uniform, lossless dielectric 
layer. Time domain scattering results and frequency domain Radar Cross Section 
computations were presented and discussed. Large peaks in the RCS were founc for the 
coated plate at large incidence angles (near grazing) due to energy being radiated from 
surface wave modes of the dielectric layer. 






5 


The next step would be to provide a rigorous analytical treatment of the problem 
of a dielectric layer on a finite sized plate (ground plane) and to derive the surface wave 
structure of the layer and the far field scattering pattern. To the best of the authors 
knowledge, no such treatment has yet been presented. Results obtained from a rigorous 
theoretical treatment would then be used for comparison with the FDTD scattering 
computations and measured data. 

V. Acknowledgement 

The authors would like to thank the NASA Ames Research Center for providing the 
necessary supercomputer resources, and Dr. Randy Jost for suggesting the problem. 

References 

[1] K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s 

equations in isotropic media." lEEETrans. Antennas P ropaqat., vol. AP-14, pp. 302- 
307, May 1966. 

[2] R. J. Luebbers et al, "A frequency dependent Finite Difference Time Domain 
formulation for dispersive materials," IEEE Trans. Electr omaan. Compat,, vol. 
EMC-32, pp. 222-227, August 1990. 

[3] R. J. Luebbers et al, "A frequency dependent Finite Difference Time Domain 
formulation for transient propagation in plasmas.' lEEE Tran s. Antennas Propaqat., 
accepted for publication. 

[4] F. P. Hunsberger, R. J. Luebbers and K. S. Kunz, "Application of the Finite- 
Difference Time-Domain method to electromagnetic scattering from 3-D chiral 
nhjflcts." Proc. IEEE AP-S Int. Svmp. . Dallas, TX, vol. 1, pp. 38-41, May 1990. 

[5] R. J. Luebbers et al, "A Finite Difference Time Domain near to far zone 
transformation." IEEE Trans. Antennas Propaqat. , accepted for publication. 

[6] C. A. Balanis, Advanced Engineering Electromagnetics, New York: Wiley, pp. 441 - 
444, 1989. 



Figure 1 . Problem geometry showing problem space size, plate size and plate position. 




THETA-POLARIZED BACKSCATTER 
3 BY 6 WAVELENGTH PLATE, THETA=0.0 


UNCOATED PLATE 
COATED PLATE 



8 10 12 14 IS IS 20 

TIME (NS) 


Figure 2. Monostatic, far zone, 0-poiarized scattered field 
for uncoated and coated 3X by 6A. plate at scattering angle 
6 =0.0 degrees. 


THETA-POLARIZED RADAR CROSS SECTION 
3 BY 6 WAVELENGTH PLATE, THETA =0.0 


UNCOATED PLATE 
COATED PUTE 


0.9 1.2 1.5 1.8 

FREQUENCY (GHZ) 


Figure 3. Monstatic Radar Cross Section versus 
frequency for an uncoated and coated 3k by 6 k plate at 
scattering angle 0 =0.0 degrees. 












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Figure 4. Monostatic, far zone, 0 -polarized scattered field 
for uncoated and coated 3X by 6A. plate at scattering angle 
0 =60.0 degrees. 



Figure 5. Monostatic Radar Cross Section versus 
frequency for an uncoated and coated 3A by 6X plate at 
scattering angle 0 =60.0 degrees. 










THETA-POLARIZED BACKSCATTER 

3 BY 6 WAVELENGTH PLATE, THETA=05.O 


40 
2 30 
> 20 

* 


UNCOATED PLATE 

- COATED PLATE 

DIELECTRIC LAYER 


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2 4 8 

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10 12 14 16 18 20 




TIME (NS) 



Figure 6. Monostatic, far zone, 9-polarized scattered field 
for uncoated and coated 3X by 6k plate and for 5 cm thick 
3A. by 6k dielectric layer at scattering angle 0=85.0 
degrees. 



Figure 7. Monostatic Radar Cross Section versus 
frequency for an uncoated and coated 3 k by 6A. plate and 
for a 5 cm thick 3k by 6A dielectric layer at scattering 
angle 0 =85.0 degrees. 



THETA-POLARIZED BACKSCATTER 
3 BY 6 WAVELENGTH PLATE, THETA=90.0 


UNCOATED PLATE 
COATED PLATE 


TIME (NS) 


Figure 8. Monostatic, far zone, 0-polarized scattered field 
for uncoated and coated 3A. by 6A plate at scattering angle 
0 =90.0 degrees. 


THETA- POLARIZED RADAR CROSS SECTION 
3 BY 6 WAVELENGTH PLATE, THETA=90.0 


COATED PLATE 


0.9 1.2 1.5 l.B 

FREQUENCY (GHZ) 


Figure 9. Monostatic Radar Cross Section versus 
frequency for a coated 3X by 6k plate at scattering angle 
9=90.0 degrees.